Recursion Relations and Qualitative Behavior of Sequences
Aaron McDonald
February 1, 2006
1
Introduction
Sequences play an important role in applied mathematics. They are often used as modeling tools for
certain kinds of populations. When sequences are used in this manner, the general behavior of the sequence
becomes an important quality that we would like to characterize. My goal today is to show you how to
make these behavior assessments. We will start by reviewing some basic sequence information, including
closed and recursion relation representations of sequences. Next, I will introduce you to a graphical method
called cobwebbing that aids in assessing the behavior of sequences defined by some recursion relation. If
time allows, we will study graphs of selected sequences to gain a basic understanding of how sequences can
behave and what this behavior may depend on.
2
Representations of Sequences
A sequence is a particular ordering of objects that is indexed by the natural numbers (N) or the non-
negative integers (Z+ + {0}). Sequences can be composed of real numbers, functions, or sets and are often
expressed as strings of these objects. Sometimes sequences can be expressed more formally by constructing
a special function and/or recursion relation. Let’s look at a couple of examples to futher illustrate these
ideas.
1
Example 1.
The sequence 1, 3, 5, 7, 9, . . . is the sequence of odd integers beginning with 1. If we
let n be any natural number and a(n) = an be the nth term of the sequence, then a(n) = an = 2n − 1 is a
function that explicitly defines this sequence. This represenation is often called the closed form representation. We could also use the recursion relation an+1 = an + 2 coupled with a1 = 1 to define this sequence
formally. Below is a side by side check that each representation independently recreates the sequence of odd
integers as I have indicated they do.
Example 2.
Sequence Term
Closed Form
Recursion Relation
an
an = 2n − 1
an+1 = an + 2, a1 = 1
a1 = 1
2(1) − 1 = 1
1
a2 = 3
2(2) − 1 = 3
1+2=3
a3 = 5
2(3) − 1 = 5
3+2=5
a4 = 7
2(4) − 1 = 7
5+2=7
.
..
..
.
..
.
The sequence x, x2 , x3 , x4 , . . . is the sequence of power functions beginning with x. This
sequence differs from the first example in that it is a sequence composed of functions rather than real
numbers. If we let n be any natural number and a(n) = an be the nth term of the sequence, then the
closed form representation of this sequence is an = xn . The recursion relation an+1 = xan coupled with
a1 = x also defines this sequence. Again, I provide a side by side check to illustrate that each representation
independently recreates the sequence of power functions.
Sequence Term
Function (Closed Form)
Recursion Relation
an
an = xn
an+1 = xan , a1 = x
a1 = x
x1
x
a2 = x2
x2
x(x) = x2
a3 = x3
x3
x(x2 ) = x3
a4 = x4
x4
x(x3 ) = x4
..
.
..
.
..
.
2
Example 3.
Let n be any natural number and a(n) = an be the nth term of a sequence. The sequence
2, 4, 8, 16, 32, . . . has the representations an = 2n and an+1 = 2an when coupled with a1 = 2.
Sequence Term
Closed Form
Recursion Relation
an
an = 2 n
an+1 = 2an , a1 = 2
a1 = 2
21 = 2
2
a2 = 4
22 = 4
2(2) = 4
a3 = 8
23 = 8
2(4) = 8
a4 = 16
24 = 16
2(8) = 16
.
..
..
.
..
.
Recursion relations are widely regarded as less satisfying representations of sequences. Why do you think
this is so? Take a closer look at the recursion relation we constructed in Example 3. What if you were asked
to compute the 10th term of this sequence given only the recursion relation an+1 = 2an coupled with a1 = 2
(and not the alternative closed form representation)? What information would you need to compute this
term?
a10 = 2a9
You would need the 9th term, and for the 9th term you would need the 8th , and so on. You must be given the
terms a1 through a9 before you could determine the value for a10 . This process could be tedious. However,
if you were given the closed form representation, an = 2n , computing the 10th term would be a piece of cake
since all you would have to do is plug n = 10 into an . Consequently, when you want to compute high or
many terms of the sequence, choosing the recursion relation over the closed form is not a good decision!
If you would like more information on sequences or their representations, I encourage you to visit the
website http://www.research.att.com/∼njas/sequences/. This website allows you to input a particular
integer sequence and it outputs known information about this sequence, including known representations
and programming notes for this sequence.
3
PROBLEM SET 1
Directions: Attempt to find a function and a recursion relation that can be used to recreate each real number
sequence.
1.
0, 1, 2, 3, 4, 5, . . .
2.
17, 17, 17, 17, 17, 17, . . .
3.
5, 3, 95 , 27
, 81 , . . .
25 125
4.
3, 9, 27, 81, 243, . . .
5.
3
3, − 32 , 34 , − 38 , 16
,...
4
6.
1, 12 , 13 , 14 , 15 , . . .
7.
1, 0, −1, 0, 1, 0, −1, 0, . . .
8.
−1, 7, 47, 223, 959, 3967, . . .
9.
1, 1, 2, 3, 5, 8, 13, . . .
5
3
Updating Functions, Initial Conditions, and Sequence Behavior
Recursion relations are often used in applied mathematics and in the sciences as representations of
sequences. Here, the sequence an is usually taken to be the number of individuals within a population at
some time step n and a recursion relation is constructed to model how this population is thought to change
between successive time steps. It is this dynamical property that makes recursion relations appealing to
modelers. These models show up in biology, anthropology, and physics. Whatever the application may be,
the purpose of constructing a model is to somehow use it to predict future states of the population. The
model may be able to predict population extinctions of massive expansions.
Do you see the problem yet? Even though recursion relations are great for modeling, they are terrible at
providing explicit information about the sequence. At best, it would be difficult to determine the long-term
trends of the population. There are a couple of options we can pursue. (1) We can compute the sequence
explicitly by iterating the recursion relation until we feel we have obtained enough terms to make accurate
predictions. This technique can be very time consuming and computationally expensive. Furthermore, if we
are not fastidious, we may make a poor or even incorrect assessment. (2) We can use the recursion relation
to find a closed form representation which we could in turn analyze. This technique can be mathematically
taxing. (3) The last avenue is to figure out a way to use the recursion relation to tease out the long-term
dynamics of the sequence without explicitly calculating the sequence or the closed form representation for
the sequence. Remember that the recursion relation dictates how the sequence changes between any two
time steps. Maybe there is a clever way to use this information to understand how the sequence behaves in
general or just for large n.
Let’s review recursion relations before we continue. Let n be any natural number and let an be the nth
term of some sequence. A recursion relation defines the process one must go through to get from any
term of the sequence, say an , to the next one, an+1 . This process may depend on any or all of the previous
sequence terms (a1 , a2 , . . . , an ) as well as the index variable n. In the language of mathematics, recursion
relations take on the form
an+1 = f (an , an−1 , . . . , a1 , n)
where f is a function that defines the before-mentioned process. The function f is sometimes called an
6
'$
&%
'$
&%
updating function as it uses known information about the sequence to produce new information about
the sequence. In the context of population modeling, f serves as a description of the forces acting on a
population between successive generations, like births and deaths.
an
f (an , an−1 , . . . , a1 , n)
-
an+1
Recursion relations alone cannot be used to represent a sequence. You must also provide some additional
information. The recursion relation an+1 = 2an requires a1 is specified while an+2 = an + an+1 requires a1
and a2 are specified. An initial condition is any information that must be specified so that a recursion
relation can be implemented.
3.1
Characterizing Long-Term Behavior
Today, we will learn a cool technique which reveals the long-term behavior of sequences with recursion
relations of the form
an+1 = f (an ).
These forms are nice in that the updating function only depends on the current state. Discrete time mathematical models in the life sciences almost always take on this form.
Cobwebbing is a graphical method that can be used to quickly and almost effortlessly reveal the behavior
of the recursion relation an+1 = f (an ) coupled with any initial condition a1 . The general idea is as follows:
1. First, to find a2 from a1 , we use the fact that a2 is the result of applying updating function to a1 .
If we were to consider a graph of the updating function, we could then easily identify a2 with the ycoordinate associated with the point on the graph of the function directly above a1 . A similiar relation
holds for all consecutive terms of the sequence.
2. Second, the axes of the above graph have special meaning. The horizontal axis represents the current
term of the sequence while the vertical axis respresents the new or updated term. To proceed from
a2 to a3 , we need to somehow get a2 from the new axis to the current axis while preserving its value.
Once we have made this move, we can identify a3 in the same manner we found a2 from a1 . What
7
happens if we reflect current terms off the diagonal line an+1 = an ? We would be moving the point
(a1 , a2 ) horizontally until it intersects the diagonal line. This means we are now at (a2 , a2 ). If we move
vertically until we intersect the updating function, we are at point (a2 , a3 ). This reflection effectively
makes the a2 transfer we wanted and it also aids in finding a3 .
Cool! Now we use this diagram to observe what happens to our sequence as we continue cobwebbing.
Cobwebbing
5
4
3
new
2
1
0
1
2
3
4
5
current
Figure 1: Graph of an+1 = f (an ) and diagonal line an+1 = an
8
Let’s practice performing the technique and reading the resulting diagram for a particular updating
function.
Example 4.
Discuss the long-term behavior of the sequence defined by an+1 = 4an − 9 with the
initial condition a1 = 4. What if we use the alternative initial conditions a1 = 2 or a1 = 3? How do these
sequences behave?
15
new 10
5
0
1
2
3
4
5
current
–5
Figure 2: Graph of an+1 = 4an − 9
9
6
7
Example 5.
Discuss the long-term behavior of the sequence defined by an+1 = a2n with the initial
condition a1 = 12 .
1
new
0.5
–1 –0.8 –0.6 –0.4 –0.2
0.2
0.4
0.6
current
–0.5
–1
Figure 3: Graph of an+1 = a2n
10
0.8
1
PROBLEM SET 2
1.
Discuss the long-term behavior of the sequence defined by an+1 = 4an − 9 with the initial condition
a1 = 3. Do the same for a1 = 0.
30
20
new
10
–6
–4
–2
2
current
4
6
–10
–20
–30
Figure 4: Graph of an+1
11
8
10
2.
Discuss the long-term behavior of the sequence defined by an+1 = an for any initial condition
a1 .
10
8
6
new
4
2
–6
–4
–2
2
4
6
current
–2
–4
–6
Figure 5: Graph of an+1
12
8
10
Discuss the long-term behavior of the sequence defined by an+1 = a2n with the initial condition
3.
a1 =
11
10 ;
Do the same for a1 = 1, a1 =
9
10 ,
a1 = − 12 , and a1 = −3.
15
new 10
5
0
1
2
3
4
current
–5
Figure 6: Graph of an+1
13
5
6
7
4.
Exploratory Exercise. Discuss the long-term behavior of sequences defined by an+1 = a3n with the
initial condition a1 . The goal here is to choose some initial conditions and see what happens. Try to expose
all types of long-term behavior this recrusion relation exhibits (depending on initial conditions of course).
2
new
1
–1
–0.5
0.5
current
–1
–2
Figure 7: Graph of an+1
14
1
5.
an
Exploratory Exercise. Do the same thing for an+1 = an e1− 10 with the initial condition a1 .
15
6
a
n
bn
cn
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30
n
Figure 8: Graph of an+1 = 4an − 9
16
1000
800
dn
en
600
400
200
0
−200
−400
−600
−800
−1000
0
5
10
15
n
20
Figure 9: Graph of an+1 = 4an − 9
17
25
30
2
gn
1.5
1
0.5
0
−0.5
−1
−1.5
−2
0
2
4
6
8
10
12
14
16
18
20
22
n
Figure 10: Graph of an+1 = 4an − 9
18
24
26
28
30
4
gn
h
n
i
n
j
3
n
2
1
0
−1
−2
−3
1
2
3
4
5
6
7
n
Figure 11: Graph of an+1 = 4an − 9
19
8
9
10